Advertisement

Advertisement

Logical armour: A primer in mathematical self-defence

Decisions about armouring US war planes were easy to get wrong (Image&colon; Photoquest/Getty Images)

Can you spot a fake health crisis? Are politicians bamboozling you with numbers? Mathematics can open your eyes, says Jordan Ellenberg in How Not to Be Wrong

IN 2008, about 66 per cent of the US population was overweight. Even more alarming than the number of fat Americans was the rate at which obesity was rising, more than 25 per cent over the previous three decades. These statistics led researchers at the Johns Hopkins Bloomberg School of Public Health to ask&colon; in what year will all Americans be overweight? The answer, published in Obesity, was 2048. It spurred a media frenzy, with ABC News announcing an “obesity apocalypse”.

Jordan Ellenberg was and is not convinced. He has no background in public health, but he is a mathematics professor at the University of Wisconsin-Madison, with plenty of experience of the linear regression used to produce the results. Linear regression reveals trends and can project them into the future by drawing a straight line through a cloud of data points. The technique is “versatile, scalable, and as easy to execute as clicking a button on your spreadsheet”, writes Ellenberg in How Not to Be Wrong. “That’s a weakness as well as a strength. You can do linear regression without thinking about whether the phenomenon you’re modeling is actually close to linear.”

For Ellenberg, the obesity study shows how easy it is to misuse linear regression, thereby inadvertently misleading a mathematically ignorant public. Trend lines for health epidemics tend to curve as they near 100 per cent. If they didn’t, a “whopping” 109 per cent of Americans would be overweight by 2060, he notes.

While Ellenberg is a witty writer his point is not to mock inept research, but to encourage more rigorous thinking by people who cast off quadratic equations after puberty. Mathematics is “a science of not being wrong about things… Without the rigorous structure that [mathematics] provides, common sense can lead you astray.”

Without the rigorous structure that [maths] provides, common sense can lead you astray

How far astray? Ellenberg provides a fine example from the second world war, when the US military was trying to optimise the armouring of its aircraft. Examining planes returning from enemy territory, analysts observed that the majority of bullets pierced the fuselage, with the fewest holes penetrating the engine. From the data, they reckoned that armour should be increased around the fuel tank. But they had it backwards. The reason there were so few holes around engines was because the engine was more vulnerable than the fuselage&colon; planes hit in the engine weren’t returning, and so were not counted in the studies.

The person who caught the error, and persuaded the military to armour engines, was Abraham Wald, one of the great mathematicians of his era. He knew nothing about aerial combat, but as Ellenberg recounts, that didn’t matter&colon; “A mathematician is always asking, ‘What assumptions are you making? And are they justified?’.” Without ever seeing a military plane, Wald recognised that the problem was one of survivorship bias, a phenomenon which also causes confusion as far afield as finance and parapsychology.

Ellenberg is not suggesting that Wald’s insight can be achieved by reading a 400-page book. But he proposes – convincingly – that exposure to survivorship bias and to abuses of linear regression will sensitise readers to these commonplace errors in their everyday experience.

How Not To Be Wrong is a superb primer, helping readers engage more sceptically with numbers. Ellenberg samples a dazzling range of mathematical principles, from arithmetic to non-Euclidian geometry and calculus. But the real impact of the book may yet be in politics, where distortion is most aggressive because the manipulation is usually intended.

Mathematics “gives us a way of being unsure in a principled way”, writes Ellenberg, eloquently encapsulating one of the field’s greatest strengths. If you don’t want to be wrong, know what you don’t know.